It is a known fact that the possible eigenvalues of Hermitian matrices \(X_1,\dots,X_s\) such that \(X_1+\cdots + X_s=0\) form a convex polytope characterized by a finite set of linear inequalities given inductively and called the Horn inequalities. These inequalities also give necessary and sufficient conditions on highest weights \(\lambda_1,\dots,\lambda_s\) such that the tensor product of the corresponding \(\text{GL}(r)\)-representations \(L(\lambda_1),\dots,L(\lambda_s)\) contains a nonzero invariant vector which means that \(c(\vec\lambda):=\dim(L(\lambda_1)\otimes\cdots\otimes L(\lambda_s))^{\text{GL}(r)}>0.\) When \(s=3,\) the multiplicities \(c(\vec\lambda)\) are identified with the Littlewood-Richardson coefficients. The Horn inequalities are linear, and so \(c(\vec\lambda)>0\) if and only if \(c(N\vec\lambda)>0\) for any integer \(N>0.\) This is the saturation property of \(\text{GL}(r).\) Belkale gives an alternative proof of the Horn inequalities and the saturation property by applying algebraic geometrical methods to the classical linkage between the invariant theory of \(\text{GL}(r)\) and the intersection theory of Schubert varieties of the Grassmannian. This includes a careful study of the tangent space of the intersection space, giving a geometric basis of invariants corresponding to the eigenvalues.
This extensive article intends to give a self-contained exposition of the Horn inequalities with a background in basic linear algebra and algebraic geometry. In addition, the text contains a not original study of Fulton’s conjecture, as this can be included without introducing further theory, and following Belkale’s geometric method. This states that \(c(\vec\lambda)=1\) for any integer \(N\geq 1.\)
It seems that the overall aim of the article is to provide an accessible introduction to the Horn inequalities, Schubert varieties of Grassmannians, and their interplay. Thus the article gives simple and explicit proofs of all results. In particular, the Littlewood-Richardson rule for determining \(c(\vec\lambda)\), nor the relation of a basis of invariants to the integral points of the hive polytope, is used. The text describes a basis of invariants which can be identified with the Howe-Tan-Willenbring basis constructed using determinants associated to Littlewood-Richardson tableaux.
The authors highlight that the desire for concrete approaches to questions in representation theory and algebraic geometry is motivated by recent research in computational complexity and the interest in efficient algorithms. The saturation property implies that the deciding nonvanishing of a Littlewood-Richardson coefficient can be decided in polynomial time.
The article contains an appendix with the Horn inequalities for three tensor factors and low dimensions.
The article starts by giving very nice definitions and elementary results of invariants, eigenvalues and intersections. For instance, the definition of the Kirwan cone and the Schubert cells are given.
Then definitions and a study of the theory of subspaces, flags, and positions are given. This works nicely as a reference work or a textbook and is of fundamental interest.
With the background given in the previous sections, the Horn inequalities and their link to intersection theory can be given in an explicit way by coordinates and dominance. This leads directly to a section on the sufficiency of the Horn inequalities by introducing the tangent maps in the corresponding algebraic geometry, between the different positions.
Now follows a section of invariants regarding the Horn inequalities. The Borel-Weil construction, invariants from intersecting tuples, the saturation of the Kirwan cone, and the connection of all this with intersection theory.
The final chapter is seen as bonus material; with all the preliminary theory given it is a rather small job to prove Fulton’s conjecture. So this is done in an explicit and close to elementary way, using tangent spaces and Belkale’s results.
The appendices contain examples of Horn triples and Kirwan cones in low dimension.
The authors achieve their main goal: This is a very useful text and is highly recommended as a reference for advanced work in geometry.